WEBVTT mathematics/geometry/pyo
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Welcome back to Educator.com.
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For the next lesson, we are going to go over volume of prisms and cylinders.
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**Volume** is the measure of all of the space inside the solid.
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We went over lateral area and surface area: remember, lateral area measures the area of all of the outside sides, except for the bases;
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and then, surface area would be the area of all of the sides together, including the bases.
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If you were to wrap a box, let's say, with wrapping paper, that would be surface area.
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If you were to fill the box with something (let's say water or sand or anything--just filling it up), that would be volume.
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That is going to be the volume of that box.
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Now, the box itself is a prism; prisms, remember, are solids with two congruent and parallel bases.
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They have to be opposite sides; they are two congruent and opposite bases.
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It can be any type of shape; it can be a triangle; it can be a rectangle; it can be a hexagon, pentagon...whatever it is.
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And the rest of the sides, the lateral sides (meaning the sides that are not bases) have to be rectangles; that is a prism.
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This right here is a rectangular prism, because the bases are rectangles.
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Let's say that we are going to name that the base; then, that means that the bottom is also the base.
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The volume (meaning everything inside, the space inside) is going to be the area of the base, times the height--
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the area of this right here, times the height of the prism.
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When it comes to a rectangular prism, we know that the area of the base is the length times the width.
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And then, the volume will be times the height; so length times width times height is for a rectangular prism.
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Length times width measures the area of the base; any time you have a lowercase b, that just means the segment base--
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the line segment, like maybe the measure of the bottom side or something; that is the base, lowercase b.
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When you have a capital B, that is talking about the area of the actual base, the side base.
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A capital B is the area of the base, times the height; and this is the volume.
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For rectangular prisms, it would just be the length times the width; but for any type of prism, it is going to be capital B,
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for base, the area of the base, times height; so this is the formula for the volume of a prism.
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Now, the volume of a cylinder: remember: a cylinder is like a prism in that there are two bases, opposite and congruent and parallel.
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But for a cylinder, the bases are circles: circle and circle.
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Now, to find the volume, again, we are measuring the space inside.
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So, if you were to take a can, and you were to fill it up with water or something, that would be volume.
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How much water can that can take?--that would be volume.
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The formula for this is actually the same as the formula for the prism: πr² is the area of the base;
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that is the formula for the area of a circle (which is the base), times the height.
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So, for a cylinder, volume is also capital B, for the area of the base, times the height.
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So again, prism and cylinder have the same formula for volume.
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Just think of capital B as the area of the base; whatever the base is, it is the area of that side, times the height.
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The first example: we are going to find the volume of this prism.
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Now, remember: when you have a prism, all of the lateral sides--meaning all sides that are not bases--have to be rectangular.
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If you have a prism, and you know that it is a prism for sure, then the sides that are not rectangular automatically become the bases.
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So here, I see a triangle; that would have to be the base, the side that is the base,
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which means that the side opposite has to also be the same; it has to also be congruent; if it is not, then it is not a prism.
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There is our base; to find the volume, remember, you are going to find the area of that base, times the height.
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The area of this triangle, we know, is 1/2 base times height (and this is lowercase b, times the height)--is all volume.
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This base and the height are 6 and 10; we know that those two are our measures, because they are perpendicular.
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The base and the height for the triangle have to be perpendicular, which they are; that is what this means, right here.
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So, it is 6 and 10; so 1/2 times 6 times 10, times the height of the prism (that is 4)...we just multiply all of those numbers together, and that is our volume.
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Here, 1/2 times 6 is 3, so I just cross-cancel this number out: 3 times 10 is...so all of this together is going to be...30,
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times the 4, is 120; our units are inches...for volume, it is going to be units cubed.
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For area, we know that it is squared; volume is always going to be cubed.
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Here, this is the volume of this prism; again, it is the area of the base, 1/2 base times height, times the height of the prism.
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Be careful that you don't confuse this h with this h; this h is just the height for the base, for the triangle;
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and this height is the height of the actual prism.
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The next example: Find the volume of the cylinder.
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Again, the formula is the same: capital B (for area of the base), times the height.
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In this case, the base is a circle; the area of a circle is π (oh, that is not π) r², times the height.
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πr...the radius is 4, squared, times 10 for my height...just to solve this out, to simplify this,
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it is going to be 4², which is 16, times 10; that is 160π.
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To turn that into a decimal, to actually multiply this out, 160 times π, you can go ahead and use your calculator.
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I have mine on my screen; so it is 160 times π; so my volume becomes 502.65; my units are centimeters, and then volume is cubed.
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So, there is the volume of my cylinder.
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The next example: A regular hexagonal prism has a length of 20 feet, and a base with sides 4 feet long; find the volume of the prism.
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Let's see, a regular hexagonal prism: "hexagon" means 6 sides, so that means the base of my prism is going to be a hexagon.
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Let me try drawing this as best I can--something like that.
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It has a length of 20 feet and a base (that is this hexagon right here) with sides four feet long; so each of these is four feet.
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Make sure that you remember that it is regular, so it has to be equilateral and equiangular.
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That means that each of these sides is going to be 4 feet long; find the volume of this prism.
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Back to the formula: volume equals the area of the base, times height.
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Now, to find the area of this base, it is a hexagon; so remember: to find the area of a regular polygon...let me just review over that quickly.
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If I have a regular hexagon (that doesn't really look regular, but let's just say it is), it is as if I take this hexagon,
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and I break it up into congruent triangles: I have 1, 2, 3, 4, 5, 6 triangles.
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The area of each of these triangles is 1/2 base times height; so if this is the base, and this is the height (that is for this triangle),
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and then I have 6 of them--so then, here is 1, 2, 3, 4, 5, 6--it is 1/2 base times height; multiply that by 6.
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Well, let me just use a different color, just to show you that this is on-the-side review.
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If I take the base (this is the base), and I multiply it by the 6, well, if this is the base, there are 6 sides.
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Then, my base and my 6 become the perimeter; and then, my height of the triangle, this right here, is what is called the apothem.
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And then, my 1/2...so then, the area of this whole thing, which is 1/2 base times height, times 6,
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because there are 6 triangles, becomes 1/2 times the perimeter times the apothem.
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So, this base is 1/2 times the perimeter times the apothem, and then times this height right here, the height of the prism.
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Let's see, my perimeter is going to be (since it is a regular hexagon) four times...I have 6 sides, so...4 times 6 is 24;
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and then, my apothem, remember, is from the center to the midpoint of one of the sides; so that is perpendicular.
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That is my apothem; now, I don't know what my apothem is, so I have to solve for it.
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So, I am going to make a right triangle; a right triangle is always the best way to find unknown measures.
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I am just going to draw that triangle a little bigger, just so that you can see.
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My apothem (that is this right here): if this whole side is 4, then I know that this right here has to be 2; it is half.
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And then, since that is all I have to work with, let's see: if I had this measure,
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I could use the Pythagorean theorem, a² + b² = c²; but I don't.
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If you did have that, then you could use the Pythagorean theorem.
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What I can do: I know that from here all the way around, passing through all 6 triangles, it is going to have a measure of 360 degrees.
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I just want to find the measure of this angle right here: 360 divided by each of the 6 triangles is going to be 60 degrees.
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That means that this right here, just one triangle, is going to have a measure of 60;
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it is going to have a measure of 60; this is 60; 60; and 60, and so on.
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If from here, all of this is 60, then I know that half of this right here has to be 30 degrees.
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If this is 30, and this is 90, then this has to be 60; and that is 0...60 degrees.
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So then, this becomes a 30-60-90 degree triangle; and again, I found out that this is 30
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by taking my 360, which is the whole thing, and dividing it by 6 (because there are 6 triangles).
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So then, this triangle and this triangle, all 6 of them--each of them is going to have an angle measure of 60 degrees.
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So, this is 60; but then I have to cut it in half again, so then, that is 30.
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This becomes a special right triangle--remember special right triangles?--it is a 30-60-90 degree triangle.
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The side opposite the 30 degrees is going to be n; the side opposite the 60 is n√3; the side opposite the 90 is 2n.
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n is just the variable for the special right triangles; that is what I am going to use.
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Now, what side is given to me? I have a 2 here--that is the side opposite the 30.
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That is this right here; so I am going to make those two equal to each other: n = 2.
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Well, if n equals 2, then what is the side opposite the 60? That is going to be 2√3.
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Then, the side opposite the 90 is going to be 2 times n, which is 4.
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OK, that means that the side opposite the 60 is 2√3; the side opposite the 90 is 4.
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My apothem, which is this side right here, is going to be 2√3; my height of the prism (because we are back to this volume formula--
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not the height of the triangle, but the height of the prism) is 20 feet.
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So again, it is 1/2 times the perimeter, times the apothem, times the height.
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To find the perimeter, remember (and all of this, 1/2Pa--that is for the area of the base): you are going to take the 4,
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and multiply it by all of the sides; that is 24; to find the apothem, you just use special right triangles.
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This is 2, so then this is 2√3, and then multiply that by 20; so I am going to have...
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and then, you can just use your calculator for that...times that number, times...and I get 831.38 feet cubed for volume.
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So, the volume of this prism is that right there.
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And then, the fourth example: Find the volume of the solid.
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For this, we have two prisms stacked like that; so to find the volume of this whole thing, I need to find the volume of this prism,
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the one on the bottom; find the volume of this top prism; and then add them together.
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I am going to say that this is prism #1 here; this is #1, and this is #2.
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#1 is going to be...volume equals capital B (for the area of the base), times the height.
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The base...this is a rectangular prism, so it is 10 times 10; the length times the width is 10 times 10
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(because it is the same; they are congruent) times the height, which is 4.
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So, the volume of this first prism, prism #1, is 100 times 4, which is 400 meters cubed.
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Then, I have to find the volume of prism #2; it is also a rectangular prism, so length times width times the height.
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The length and the width are congruent, so this is going to be 6 times 6; and the height is 2.
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I know that 6 times 6 is 36, times 2 is 72; that is meters cubed.
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The volume of this solid is going to be prism 1, plus prism 2.
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Prism 1 is 400 meters cubed, plus the 72 meters cubed, which is going to be 472 meters cubed.
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Make sure that you add the two volumes together; you are not multiplying them,
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because it is the volume of that whole thing, plus the volume of this whole thing.
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You are adding them together--the total is the sum of them.
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That is it for this lesson; thank you for watching Educator.com.